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Symmetric Generation and Existence of J3:2, the Automorphism Group of the Third Janko Group

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Symmetric Generation and Existence of J3:2, the Automorphism Group of the Third Janko Group View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 304 (2006) 256–270 www.elsevier.com/locate/jalgebra Symmetric generation and existence of J3:2, the automorphism group of the third Janko group J.D. Bradley, R.T. Curtis ∗ School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Received 27 June 2005 Available online 28 February 2006 Communicated by Jan Saxl Abstract We make use of the primitive permutation action of degree 120 of the automorphism group of L2(16), the special linear group of degree 2 over the field of order 16, to give an elementary definition of the automorphism group of the third Janko group J3 and to prove its existence. This definition enables us to construct the 6156-point graph which is preserved by J3:2, to obtain the order of J3 and to prove its simplicity. A presentation for J3:2 follows from our definition, which also provides a concise notation for the elements of the group. We use this notation to give a representative of each of the conjugacy classes of J3:2. © 2006 Elsevier Inc. All rights reserved. 1. Introduction The group J3 was predicted by Janko in a paper in 1968 [10]. He considered a finite simple group with a single class of involutions in which the centraliser of an involution 1+4 was isomorphic to 2 :A5. The following year Higman and McKay, using an observation of Thompson that J3 must contain a subgroup isomorphic to L2(16):2, constructed the group on a computer [9]. Weiss constructed J3 as the automorphism group of a graph with 17442 vertices [12]. This graph has diameter 10 and has a rather complicated structure. * Corresponding author. E-mail address: [email protected] (R.T. Curtis). 0021-8693/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2005.09.046 J.D. Bradley, R.T. Curtis / Journal of Algebra 304 (2006) 256–270 257 Kleidman and Wilson realised an embedding of J3 into E6(4) [11], which was later proved · without computers by Aschbacher [1]. More recently Baumeister has constructed 3 J3 in terms of an amalgam [2]. ∼ In this paper we give a concise definition of Aut(J3) = J3:2. The minimum degree per- mutation representation for J3 is 6156, and as complex matrices the minimum degree is 85 (170 for J3:2). Our construction gives a much more concise way of recording elements of J3. · Now J3 possesses a triple cover 3 J3 which has a 9-dimensional irreducible representa- tion over F4, the field of order 4, see the ATLAS [6]. Our definition of Aut(J3) is shown to define either a group of order 100465920 containing a simple subgroup to index 2, or the cyclic group of order 2. The 9-dimensional representation is used to show that the former, in fact, holds. ∼ We consider the (primitive) action of Aut(L2(16)) = L2(16):4 on 120 letters. We take 120 = | 2 = 2 t1,...,t120 ti 1 , the free product of 120 copies of the cyclic group of order 2, 120 and construct a semi-direct product of the form P = 2 :(L2(16):4); as in Curtis [7,8], we refer to such an infinite group as a progenitor. The second author argued that J3:2is a homomorphic image of P as follows. The normaliser of a Sylow 17-subgroup of J3:2 ∼ is of the form 17:8 × 2; the subgroup N = L2(16):4ofJ3:2 contains a maximal subgroup isomorphic to 17:8 which must be contained in this normaliser, and so centralises an invo- 17×16×15×4 = lution. Under conjugation by N this involution thus has 17×8 120 images which, together with the maximal subgroup N, generate J3:2. Indeed, the set of 120 involutions generates a subgroup of J3:2 which is normalised by the whole group and which is not contained in J3; thus it forms a generating set for J3:2. Now elements of P can be written, essentially uniquely, in the form πw where π is a permutation in N and w isawordinthe 120 symmetric generators; thus any relator by which we might factor P has this form. We produce below an easily described relator, found by John Bray, such that P ∼ G = = J3:2, πwP where πwP denotes the normal closure of the subgroup πw. ∼ The subgroup N = L2(16):4 has index 6156 in J3:2. We are able to construct a graph of rank 7 on 6156 vertices by joining a coset Nw of N to the 120 cosets Nwti , assuming the above relator holds (and, of course, all of its conjugates by permutations of N). This shows that G has order at most 6156 ×|L2(16):4|. It is shown that this graph has diameter 3, so we can represent elements of J3:2 by an element of L2(16):4 followed by a word in the 120 symmetric generators of length at most 3. 2. L2(16):4 and its action on 120 letters In what follows we shall be defining and constructing a group which contains a set T of ∼ 120 involutions whose set normaliser N is isomorphic to Aut(L2(16)) = L2(16):4 acting primitively on T by conjugation. We shall need to understand this 120 point action in some detail. 258 J.D. Bradley, R.T. Curtis / Journal of Algebra 304 (2006) 256–270 Table 1 The permutation character of degree 120 Centraliser 16320 64 60 30 15 17 17 Class 1A 2A 3A 5AB 15ABCD 17ABCD 17EFGH Fixed points 120 8 0 0 0 1 1 Shape 1120 18.256 340 524 158 1.177 1.177 240 16 12 10 24 24 8 8 12 12 2B 4A 6A 10AB 4B 4C 8A 8B 12A 12B 04 00 002 2 0 0 260 14.22.428 620 1012 430 430 12.2.4.814 12.2.4.814 1210 1210 To obtain a copy of the field of order 16 we extend the prime field Z2 by a root, ν say, 4 ∼ 4 of the irreducible quartic x + x + 1. Thus we let F = Z2(ν) = Z2[x]/(x + x + 1).Welet σ be the field automorphism of order 4 which maps each element of F to its square. We ∼ represent each element of N = L2(16):4bya2× 2 matrix of determinant 1 with entries from F , followed by a power of σ . Let the set of Sylow 17-subgroups of N be denoted by S. Then |S|=120 and N acts primitively, by conjugation, on S. In order to obtain a convenient notation for the members of S we proceed as follows. For each H ∈ S we choose the unique element πH ∈ H which, when it acts on the projective line P1(16), maps π2 ∞ to 0. We label H by the ordered pair (0πH , 0 H ). The group N is sharply 3-transitive on P1(16), so this pair defines the subgroup uniquely. The normaliser of one such Sylow 17-subgroup has shape 17:8. Inducing the trivial character of this subgroup up to N (or equivalently, writing down for an element of each conjugacy class of N the number of Sylow 17-subgroups it normalises) we obtain the permutation character shown in the Table 1, where the classes are labelled as in the AT- LAS [6]. This permutation character has norm 4 and using it we may readily write down the cycle shapes of elements in the 120-point action. In particular we see from Table 1 that elements of order 8 have cycle shape 12.2.4.814 and that all fixed point free elements are semi-regular, i.e. their cycles all have the same length. Since elements of order 8 have two fixed points we see that one of the nontrivial orbits of the point stabiliser has length 17, and since the other two have lengths of the form 2a.17 they must contain 34 and 68 points. We say that 2 points are α-joined, β-joined or γ -joined according to whether they lie in the 17-, 34- or 68-orbits of one another’s stabiliser. In Fig. 1 we give a diagram of the resulting α-graph, indicating the orbit lengths of the stabiliser of a point in each of the four suborbits. Thus, for instance the stabiliser of two points x and y which are β-joined to one another has an orbit of length 4 on points α-joined to both x and y, three orbits of length 4 on points γ -joined to x and α-joined to y, and a unique point β-joined to x and α-joined to y. Now an element of order 8 lying in our point stabiliser must, by simple arithmetic, act on these suborbits as 1 + 1 + 82 + 2 + 84 + 4 + 88 and so points in its 2-orbit must be β-joined to its fixed points and points in its 4-orbit must be γ -joined to its fixed points. This shows us that the subgraph on {a,b,c,d,e,f,g,h}, J.D. Bradley, R.T. Curtis / Journal of Algebra 304 (2006) 256–270 259 1 34 4 43 8 17 1 1 17 8 23 2 68 1 + 24 Fig. 1. The graph on 120 letters. ac eg bd fh Fig. 2. The fix of an involution. the fixed point set of an involution, has form consisting of 4 α-edges ab,cd,ef and gh and otherwise points in the same half are β-joined whilst those in opposite halves are γ -joined, see Fig.
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